On Bernstein type quantitative estimates for Ornstein non-inequalities

TL;DR

This paper establishes lower bounds on Bernstein-type constants for certain inequalities involving derivatives of trigonometric polynomials on the torus, revealing a logarithmic growth rate depending on geometric properties.

Contribution

It provides the first quantitative estimates for Ornstein non-inequalities, linking geometric configurations of derivatives to growth rates of constants.

Findings

Lower bounds grow at least as a power of log N

Constants depend on geometric properties of derivative orders

Results apply to multi-dimensional torus setting

Abstract

For the sequence of multi-indexes $\{\alpha_i\}_{i=1}^{m}$ and $\beta$ we study the inequality \[ \|D^{\beta} f\|_{L_1(\mathbb{T}^d)}\leq K_N \sum_{j= 1}^{m} \|D^{\alpha_j}f\|_{L_1(\mathbb{T}^d)}, \] where $f$ is a trigonometric polynomial of degree at most $N$ on $d$-dimensional torus. Assuming some natural geometric property of the set $\{\alpha_j\}\cup\{\beta\}$ we show that \[ K_{N}\geq C \left(\ln N\right)^{\phi}, \] where $\phi<1$ depends only on the set $\{\alpha_j\}\cup\{\beta\}$.